Connection to K-theoretic classification schemes.
The existence of a mass term gapping anomalous states related by \(n\) -fold rotation symmetry can be determined from the existence of a strong second-order phase protected by \(n\) -fold rotation symmetry in the respective symmetry class and dimension as determined from \(K\) -theoretic methods (see Refs. [1]}, [2]}, [3]}, [4]}). This follows as it has been shown in these articles that a strong second-order topological phase can be deformed into symmetry related \(d-1\)  dimensional building blocks decorated with \(d-1\)  dimensional first order topological phases. This limit is identical to the topological crystal limit locally around a rotation axis. This draws a one-to-one correspondence between the existence of mass terms hybridizing symmetry related \(d-2\)  dimensional anomalous states at a rotation axis and the existence of a second-order topological phase protected by the same rotation symmetry.
